# More on Interest Rates and Inflation

In my previous post, I argued that New Keynesians do not understand monetary policy. The New Keynesian model predicts that if one pegs the interest rate, then increases in the interest rate cause increases in the rate of inflation. New Keynesians, however, argue that in order to stimulate economic activity and increase inflation, the central bank needs to reduce nominal interest rates. Thus, I argued that either their policy advice is wrong and their model is correct or their model is correct and their policy advice is wrong.

In this post I would like to argue that the problem with the New Keynesian view is that it completely ignores money. By ignoring money, the New Keynesians assume that the central bank need only use the nominal interest rate as a monetary policy tool. Previous generations of monetary economists argued that interest rate policies were not plausible. An earlier generation of economists, such as Milton Friedman and Jurg Niehans, argued that pegging the nominal interest rate would cause ever-increasing inflation. A later generation of economists, such as Thomas Sargent and Neil Wallace, argued that pegging the interest rate would lead to multiple equilibria. The logic of these claims are the same, but the primary difference in their analysis is the treatment of expectations. Under Friedman’s and Niehan’s formulations, expectations were backward-looking. For Sargent and Wallace, agents were assumed to have rational expectations and are therefore forward-looking. The reason for the different predictions are the differences in assumptions about expectations, but the perils of interest rate targeting remain.

New Keynesians, however, seem to believe that they have solved this problem with the Taylor principle, which states that when inflation rises by 1%, the central bank should increase the nominal interest rate by more than 1%. The intuition behind this is that by raising the nominal interest rate by more than the rate of inflation, the central bank is increasing the real interest rate, which reduces demand and thereby inflationary pressures. Using the Taylor principle, New Keynesians were able to show that they could get a unique equilibrium using an interest rate target. In addition, their model showed that they could do this all without money.

Despite the ability of New Keynesians to get a unique equilibrium with an interest rate target and no reference to money, remained critics of the practical application of the model. A number of critics, for example, pointed out that the way in which the Federal Reserve adjusts its own interest rate target, the federal funds rate, is through open market operations. The Federal Reserve increases the supply of bank reserve by making open market purchases and this causes the federal funds rate to fall. Thus, the critics argued, that even when the Fed uses the federal funds rate as its preferred measure of communicating policy, the Fed’s actual policy instrument is the monetary base.

The New Keynesians, however, countered that they didn’t need to use open market operations to target the interest rate. For example, Michael Woodford spends a considerable part of the introduction to his textbook on monetary economics to explaining the channel system for interest rates. If the central bank sets a discount rate for borrowing and promises to have a perfectly elastic supply at that rate and if they promise to pay a rate of interest on deposits, then by choosing a narrow enough channel, they can set their policy rate in this channel. In addition, all they need to do to adjust their policy rate is to adjust the discount rate and the interest rate paid on reserves. The policy rate will then rise or fall in conjunction with the changes in these rates. Thus, the New Keynesians argued that they didn’t need to worry about money in theory or in practice because they could set their policy rate without money and their model showed that they could get a determinate equilibrium by applying the Taylor principle.

Nonetheless, what I would like to argue is that their ignorance of money has led them astray. By ignoring money, the New Keynesians have confused cause and effect. This confusion has led them to believe that they know something about how interest rate policy should work, but they have never stopped to think about how interest rate policy works when the central bank adjusts the nominal interest rates in a channel system versus how interest rate policy works when the central bank adjust the nominal interest rate using open market operations.

The standard story about the impact of monetary policy on nominal interest rates as follows. Suppose that the central bank increases the growth rate of money. Assuming that we are initially in equilibrium, this increase in the growth rate of money initially increases the nominal supply of money such that the real value of money balances is greater than individuals desire to hold. As a result, the nominal interest rate must fall to induce individuals to hold these higher money balances. This decline in the interest rate is referred to as the liquidity effect. Over time, however, this increase in money growth causes inflation to increase. Thus, as inflation expectations rise, this puts upward pressure on the nominal interest rate. This is referred to as the Fisher effect.

Keynesians tend to focus on the liquidity effect as the source of the transmission of monetary policy. For example, this decline in interest rates (coupled with the assumption of sticky prices) should increase both consumption and investment as the real interest rate also declines. Thus, the New Keynesians view the nominal interest rate as a tool in and of itself. However, New Keynesians are confusing cause and effect. It is called the liquidity effect because the decline in interest rates is caused by a change in the money supply.

This distinction is eminently important when one considers the implications of paying interest on reserves. For example, consider the world without interest on reserves. The standard story about the market for reserves is that the Federal Reserve determines the supply of reserves and therefore the supply curve for reserves is vertical. The demand for bank reserves is decreasing in the nominal interest rate. Thus, an increase in the quantity of bank reserves shifts the supply curve to the right. At the old interest rate, banks want to hold less reserves than are supplied. Thus, the interest rate has to adjust downward to get us back to equilibrium. This is shown in the figure below.

Now consider the effects of this change within the following basic framework. Recall that the equation of exchange can be expressed in logarithms as

$m + v = p + y$

where m is a broad measure of the money supply, v is velocity, p is the price level, and y is real income. The equation of exchange is identically true. However, we can use the equation of exchange to discuss the implications of the quantity theory of money if we have some theory of velocity. Suppose that we adopt the standard Keynesian liquidity preference view of money demand:

$m - p = \alpha y - \beta i$

Thus, the demand for real money balances is increasing in real income and decreasing in the nominal rate of interest. Assume for a moment that $\alpha = 1$. It is straightforward to show that we can solve this expression for the demand for money to get a theory of velocity:

$v = \beta i$

Thus, velocity is an increasing function of the nominal interest rate. Substituting this into the equation of exchange, we have:

$m + \beta i = p + y$

Now consider the effects of a change in the money supply. As illustrated in the figure above, the increase in the money supply causes the interest rate to decline. This means that when the money supply increases, velocity declines. However, the interest elasticity of velocity is often estimated to be rather small. The initial effect of the increase in the money supply is that the nominal interest rate to fall and nominal spending to rise. The decline in the nominal interest rate is an effect of the change in the money supply, but note that it is not the cause of the change in nominal spending.

However, now consider what happens if the central bank pays interest on reserves. We now have to re-draw our market for reserves. Specifically, the demand for reserves will now be kinked. This is because interest on reserves imposes a floor on the interest rate in this market. Nonetheless, since the interest rate on reserves is set by central bank policy, they can adjust this rate without open market operations. If the quantity of reserves in the banking system is sufficiently large, an increase in the interest rate paid on reserves will not have an effect on the equilibrium quantity of reserves. To see this, consider our modified market for reserves presented below.

[Updated: Scott Sumner made clear to me in the comments that what initially followed this point was hard to understand. As a result, it has been edited to be more clear. –JH]

Thus, let’s accept the New Keynesian idea that the change in the interest rate on reserves can be done in the absence of a change in broad measures of the money supply (we’ll return to this point later). Recall that the Keynesian liquidity preference of the money supply implies that the income velocity of the money supply is $\beta i$. Then, returning to our equation of exchange we have:

$m + \beta i = p + y$

Now consider what happens if the central bank increases the interest rate paid on reserves, holding the money supply constant. According to the expression above nominal spending increases when the interest rate rises. To understand why, consider that the interest rate on excess reserves reduces the demand for currency. Since the supply of currency doesn’t change, individuals increase spending to alleviate attempt to alleviate this excess supply. However, the excess supply is alleviated only once prices have risen such that real currency balances are back in equilibrium. It is important to note that this increase in velocity results only because individuals are pushed off of their demand curve (i.e. the individual is temporarily off of their currency demand curve). This disequilibrium is a characteristic of most supply and demand analyses. The supply and demand for reserves, however, is given by a cross. There is never a point of disequilibrium. As a result, the velocity of reserves is unchanged.

Note here that this result comes entirely from adopting New Keynesian-type assumptions in a static model. There is no role for expectations and I haven’t done any fancy tricks with difference equations.

So is the New Keynesian model wrong and, if so, where does it go wrong? My own view is that the New Keynesian model is likely incorrect. And the New Keynesian model is likely incorrect for two reasons. First, the New Keynesians have no theory of the money supply process. In other words, how do changes in the interest rate paid on reserves affect the supply of broader measures of the money supply? The New Keynesian view seems to be that since the change in the rate doesn’t affect bank reserves, then it doesn’t affect the money supply, more generally. I think this needs to be fleshed out in a formal model rather than in a static partial equilibrium graph of the market for reserves.

Second, the typical New Keynesian liquidity preference view of money demand is wrong (this is by the way, the New Keynesian view, you can find it in Gali’s book on the NK model). The money demand equation should be a function of the price of money, not “the” nominal interest rate (see my recent post on the price of money here). This is important because the implications of paying a higher rate of interest on reserves for the opportunity cost of holding broad measures of money is not obvious.

Taking these points slightly more seriously, we could re-write the equation of exchange as

$mm(MB) + v(\phi) = p + y$

where $mm$ is the money multiplier, $\phi$ is the price of money, and $MB$ is the monetary base. A model of broad money demand would imply that a higher interest rate on reserves would increase the opportunity cost of holding currency and reduce the opportunity cost of holding deposits. The former would increase $mm$ whereas the latter would reduce $v$. Thus, the effect on inflation would depend on the relative magnitude of this effect. If the effects were of equal magnitude in absolute value, then the inflation rate wouldn’t change. This is precisely the outcome shown in Peter Ireland’s model of interest on reserves which appends the standard NK model to have a role for broad money:

### 4 responses to “More on Interest Rates and Inflation”

1. Josh, Interesting post but you lost me near the end. When hter eis no IOR, a higher interest rate causes base velocity to rise. But when the Fed uses IOR as a policy tool, a higher interest rate causes the velocity of base money to fall. And yet you seem to indicate that Bi would rise.

Using broad money doesn’t really change anything, except that now the multiplier also adjusts.

2. Sorry for the typo, I meant “When there is no IOR”

3. You are correct, the end of that post needs to be re-written. If I have a chance today, I will do so. Unfortunately, I was trying to make two different points and they ran together. Here are the two points:

1. Suppose the Fed increased the interest rate on reserves. NKs argue that this would increase the demand for reserves, but since the demand curve is horizontal at the IOR and the supply curve is vertical at the quantity of reserves then there is no change in the monetary base. However, the increase in the interest rate reduces the demand for currency and therefore money demand falls (velocity rises). If M is constant, then whether nominal spending rises depends on the interest elasticities of the demand for currency and the supply for reserves (since a change in the interest rate moves us along the demand curve for currency and along the supply curve for reserves). Since the supply of reserves is assumed to be vertical, the interest elasticity of supply is zero, the currency effect dominates.

2. The problem with the NK approach is that it ignores any useful treatment of the demand for money. Suppose instead that we write the equation of exchange like this:

mm(MB) + v = p + y

where mm is the money multiplier and MB is the monetary base. An increase in the interest rate on reserves should cause other interest rates to increase or at least cause the demand for currency to decline. The former causes an increase in the demand for broad money balances and therefore v should decline. However, the reduction in the demand for currency causes the money multiplier to increase. Thus, the question is whether the increase in mm(MB) is greater than, less than, or equal to the decrease in v. NKs don’t have a consistent answer to this.

In fact, if you look at Peter Ireland’s model of IOR in a NK model, inflation does not respond at all to a change in IOR. This is potentially problematic considering people like Bernanke are on record saying that they’ll increase IOR if inflation rises.